96
$\begingroup$

I'm in my first semester of college going for a math major and it's pretty great. I'm doing well, however, there seems to be huge gap between how difficult /complex an idea is and how convoluted it is presented.

Let me make an example: In Analysis we discussed the Bolzano Weierstrass theorem and one of the lemmas showed that every sequence in $\mathbb{R}$ has a monotone subsequence. The idea behind the proof with the maximum spots ( speaking colloquially here ) is super simple and pretty elegant if you asked me, but I spent a significant amount of time trying to understand the notation of the professor until I went to this site to read a "proper explanation" of the proof, which had much simpler notation in it.

Extracting the idea of the proof took me lots of time because of the strange notation, but once you understand what is going on, it is really easy. Most of the time spent studying lectures is about digging through the formalities.

Do I just have to spend more time really going through all the formal details of a proof to become accustomed to that formality? Or do more advanced mathematicians also struggle to extract the ideas from the notation?

I'd assume there will come a point, where the idea itself is the most complex part, so I do not want to get stuck at the notation, when that happens.

( Proof Verification - Every sequence in $\Bbb R$ contains a monotone sub-sequence if you are interested )

$\endgroup$
  • 90
    $\begingroup$ In my experience, it really is true that one of the main difficulties in learning math is that often things are not explained very well. Sometimes you finally find an explanation by someone who has grokked the idea, and who can show how simple the idea is, and suddenly you realize it should not have been so hard to understand. While new research contributes to civilization's mathematical progress, another important aspect of mathematical progress is figuring out how to explain things so clearly that people can grok a vast amount of knowledge quickly and easily. $\endgroup$ – littleO Nov 20 '16 at 15:30
  • 4
    $\begingroup$ Are you talking about notation like $\forall n \in \mathcal S \exists m \implies P \iff$ etc.? $\endgroup$ – Robert Soupe Nov 20 '16 at 16:21
  • 7
    $\begingroup$ @littleO: Your comment should have been an answer, because that is really true of a lot of textbooks and teaching material, which simply don't teach well. There are also some very clever people who are not good at conveying what they know to others. $\endgroup$ – user21820 Nov 20 '16 at 17:11
  • 4
    $\begingroup$ An excellent exercise is to take a book-concise proof/demonstration and write it out fully-expanded to the extent that you can understand every line (as you've written it) on sight. The stuff I've done that for is the stuff I understand and remember the best. Do it for stuff you're most interested in (time permitting). $\endgroup$ – Daniel R. Collins Nov 20 '16 at 19:50
  • 4
    $\begingroup$ It's worth noting that different STEM practitioners also use different notation, with overlapping typography. Linear algebra is rife with different notations. So it's not all uniformly difficult, but neither is it consistent. $\endgroup$ – bright-star Nov 20 '16 at 20:51

10 Answers 10

128
$\begingroup$

As others have pointed out, it gets much better if that's your first semester. But in my experience, there is not much relief between, say, years 2 and 4 of your studies. Sure, you get more mature, but the material gets more difficult too.

To address your question whether "more advanced mathematicians also struggle to extract the ideas from the notation", I'd like to quote V.I. Arnold, since I think it's exactly in the spirit of your frustration.

It is almost impossible for me to read contemporary mathematicians who, instead of saying "Petya washed his hands," write simply: "There is a $t_1<0$ such that the image of $t_1$ under the natural mapping $t_1 \mapsto {\rm Petya}(t_1)$ belongs to the set of dirty hands, and a $t_2$, $t_1<t_2 \leq 0$, such that the image of $t_2$ under the above-mentioned mapping belongs to the complement of the set defined in the preceding sentence.''

The trade-off is clear: without rigor math would've been quite a mess. But if rigor is the only way math gets communicated to someone, this person simply won't have time to get far in math.

$\endgroup$
  • 35
    $\begingroup$ Great points, and fun example ;) $\endgroup$ – String Nov 20 '16 at 17:49
  • 22
    $\begingroup$ Especially since that version implicitly relies on the Intermediate Value Theorem to say what it is supposed to mean. $\endgroup$ – Toby Bartels Nov 21 '16 at 19:56
  • $\begingroup$ This quote does not seem right, can you provide a source? $\endgroup$ – Jimmy R Nov 26 '16 at 22:21
  • 1
    $\begingroup$ @JimmyR: "Conversation with Vladimir Igorevich Arnol’d" (Arnold interviewed by Smilka Zdravkovska), The Mathematical Intelligencer, December 1987, Volume 9, Issue 4, p. 30. $\endgroup$ – Leo Nov 27 '16 at 9:09
  • $\begingroup$ The most incredible thing about this quote (assuming it's accurate, since I can't access the source) is the word 'simply'. It is, of course, it is far easier to read the first version, but if you try to write the first version, you'll see what Arnold was getting at... $\endgroup$ – Eric Stucky May 11 '17 at 22:54
30
$\begingroup$

The question you need to ask yourself is: "If a lecture contained only ideas and intuitions without rigorous formalities, would I be able to write formal proofs on my own?"

If the answer is "yes", then congratulations, you are very promising young mathematician. If it is not the case, you are in the same boat as most of us were when we began studying mathematics (you might still be promising young mathematician, though).

The thing is, formal language is a necessity when we want to clearly state our ideas, check if they are correct and share them with others. Sure, two experts in a field might not need to be very formal when communicating with each other and still understand perfectly what they mean and you listening on the conversation might not understand a single thing they said. If you were interested, you'd want them to tell you the appropriate formal definitions and theorems involved. You might want to know about particular details in some proof and how does one reach desired conclusions. This would not be possible without formal context. Then again, even two experts might find themselves in disagreement in which case they would go back to being formal to clarify things to their satisfaction. Lack of rigor can obscure subtle errors and we've all fallen prey to it at some point. Hence, it is essential for any mathematician to be able to read formal proofs and to write their own.

The beginning courses in mathematics ought to teach you rigor and formal language in order to prepare you for understanding advanced topics. You might struggle now, but when you grasp it, you will be very thankful for it. The goal of learning proofs will shift more to understanding ideas involved and not the formal language, but at that point you should be able to write down ideas formally and judge their correctness by yourself.

To be fair, I agree that some authors are overzealous with formal language while disregarding intuition and ideas. But, this is what the faculty is for, professors and teaching assistants are there to explain and clarify. When in doubt, you should ask for their help.

$\endgroup$
  • 5
    $\begingroup$ I really like your part about promising young mathematician. Very motivating. $\endgroup$ – Kami Kaze Nov 21 '16 at 13:30
  • 1
    $\begingroup$ But why not just have both? The idea/intuition to understand what we're talking about, and what it's for, and then the formal notation. I remember a teacher in signal processing who was doing exactly this. I couldn't understand any of the equations, but I least I understood what those stuff were for. Too often teachers just teach you equations/theorems, but you don't know where they come from nor what they mean/what they're for. $\endgroup$ – user276648 Nov 23 '16 at 1:03
  • $\begingroup$ @user276648, of course, you are right. As a teacher in high school I try to do both as much as I can. But sometimes time is not permitting and one often has to compromise between quality and quantity. Good balance is a sign of a good teacher. You can't always have ideal situation. Very often complete understanding comes later (as in year or two later) when you'll find it hard to believe that you couldn't grasp it immediately. That's just how it is and all you can do is talk with others as much as you can to gain insight. $\endgroup$ – Ennar Nov 23 '16 at 10:22
15
$\begingroup$

If its your first semester taking a proof based class, then its very typical that it takes a long time to absorb proofs. The more math classes you take, the more fluent you'll become in reading and understanding proof. It gets better!

$\endgroup$
  • 2
    $\begingroup$ Additionally, the early courses are designed to get you thinking precisely and able to interpret the more formal proofs. Most of maths tends not to be performed that way. $\endgroup$ – Patrick Stevens Nov 20 '16 at 15:59
  • $\begingroup$ @PatrickStevens or you are better and it doesn't look so formal anymore. ;-) $\endgroup$ – Tacet Nov 20 '16 at 19:14
  • 1
    $\begingroup$ @PatrickStevens : Early courses are also designed to introduce you to common idioms. Like any foreign language, once you have learned common constructions, parsing and comprehension is improved. $\endgroup$ – Eric Towers Nov 21 '16 at 1:38
11
$\begingroup$

I think I disagree with @littleO's comment that

...one of the main difficulties in learning math is that often things are not explained very well...

I used to share this view. But actually I think that it is more accurate to realize that things really are just complicated and understanding them well only really ever happens in your own mind. And this takes a long time.

When you are first trying to understand some math, a certain (probably oversimplified) way of looking at it might make most sense to you (e.g. just the "main idea" presented in a non-rigorous way).

In order to master this piece of math, you might then read various different explanations, think about it in various different ways and talk to different people. As you do this, you are learning little extra facts, testing out and stretching your understanding etc. until eventually you understand this piece of math a lot better. And once you understand it well, the explanation that makes most sense to you will have changed. Then, miraculously

you finally find an explanation by someone who has grokked the idea

But in reality, the important point is that it is you who has finally "grokked" something. If you had been given this explanation in the beginning, you may not have found it so helpful because you may not have been ready to hear it.

To remark on your specific example of Bolzano-Weierstrass/existence of monotonic subsequences: The idea behind the proof is elegant and simple and maybe you really did understand it quickly. This is great!

BUT, (and I can't stress this enough):

the proof itself is not the same as the idea behind the proof; nor does it just consist of writing down the idea behind the proof

So it makes perfect sense that the proof may be fiddly and not straightforward to understand, whereas the main idea is simple.

You asked

Do I just have to spend more time really going through all the formal details of a proof to become accustomed to that formality?

One typically has to spend lots of time going through rigorous proofs in order to understand them on a detailed level. Once you build up experience and knowledge in a certain area of mathematics, this will become easier. (It will become easiest for proofs in that area, because a different area of math will have different notation and different conventions and so not all of this experience is transferable).

Or do more advanced mathematicians also struggle to extract the ideas from the notation?

It is absolutely true that more advanced mathematicians also have to spend a long time extracting ideas from proofs. But again the level of knowledge vs. the context does matter: An experienced mathematician who has been through many, many proofs in basic analysis will be able to skim such proofs for the main ideas. But when reading a new, technical research paper will still ultimately have to spend time going through the proofs carefully to slowly build up intuition.

$\endgroup$
9
$\begingroup$

"...in mathematics you don't understand things. You just get used to them."
- John von Neumann

This Question and its Answers explaining the quote seem to incorporate most of the possible answers to your question.

$\endgroup$
7
$\begingroup$

In my experience, there is a trade-off between compact notation and lengthy prose in math texts. Sometimes mathematical texts try to introduce lots and lots of notation so that what you read is more like a condensed code to be processed by a computer rather than to be read by a human. At other times, some authors use too much text AND too much notation taking the reader safely through each step in a cumbersome manner where you as a reader lose track of the goal of and idea behind the whole thing.

Getting used to interpreting notation is of course always a good thing, and in my experience this happens when you both read and write mathematical notation yourself. Sometimes writing out the compact and complicated statement from the textbook will help you grasp the idea condensed in there. Condensing lengthy prose works equally well. This way you will keep getting better.

$\endgroup$
6
$\begingroup$

It is just like learning the alphabet - it takes time, at first it seems ridiculous and incomprehensible, even if you get the gist of the story you are trying to read, it's hard to learn the specifics of the language or story or whatever. But when you do get it, it's easy to read and write, and understand what is going on, and you can't imagine ever having not understood it - it's easy too you.

Practice makes perfect - the more you practice with it, the better you'll get, and eventually it'll be hard for you to imagine having not understood all of that notation. So keep reading proofs, writing proofs, and trying to swim through the quagmire of notation - it'll get better.

$\endgroup$
2
$\begingroup$

It gets much easier to understand the notation. At a certain point you start to realize that you've seen all of the basics - the hardest notations you'll encounter are the ones introduced in the proof you're reading, and then the definition is right there for you to look at. You'll also find that understanding the topic better makes you understand the notation better - if you have a good enough sense for the situation, you can tell what the notation should be saying. You know you've started to hit this point when you start being able to correct typos without getting confused.

So my advice is to aim for an intuitive comprehension of the topic you're working with; start out by ignoring notations and just getting a feel for the situation. Once you get the hang of it, look at the notation again and see if it makes more sense.

$\endgroup$
1
$\begingroup$

The answer is that these things build one upon the next. It is pretty unusual encountering math books where the majority is presented in notation but you learn basics and work your way up, as with anything else. It becomes easier over time in other words. Trying to read a topology text without a foundations course and some analysis would be damn near impossible. So take heart in that it will get easier.

$\endgroup$
1
$\begingroup$

Others have already answered whether it gets easier during the course of your studies (it does). Suboptimal notation exists, and teachers who use suboptimal notation also exist. (Some teachers may even be deliberately trying to teach you how to chew through suboptiomal notation.)

Reading your question differently, there's also the question whether notation improves ("becomes easier") over historical time ("ever"). It does. Evolution works on ideas, including notations. But we have to be patient, it may take a century or two.

When an idea is first described, it is often done using existing notation. This makes sense, because the goal of the description is that readers understand it. New notation might discourage readers, putting the new idea at a disadvantage. Later, when the idea itself is widely understood, people may strive to streamline things, and switch to different or new, better suited notation.

Some scholars opt for a new notation immediately, from first publication, some successfully and some not so. For example, Hermann Grassmann used a new notation to introduce his ideas on multidimensional algebra. No one influential understood what he was writing (or publicly admitted to do so), so his ideas were not widely recognized. Clifford and Hamilton pursued similar ideas, with more success, possibly because they used different, more common notation (but probably more so because they were more influential to begin with).

A well-suited notation can definitely make it easier to reason about a subject. This of course applies not just to mathematics, but many subjects, for example physics, or programming. Putting a little effort in finding, adapting or crafting the "right" notation for the job is often worth it.

EDIT: In relation to learning, different courses may use competing "notations", or formalisms, to teach the same subject. For example, quantum mechanics may be taught using integrals, or using traditional matrix algebra notation, or using Bra-Ket notation. The subject is the same, but the notations look wildly different. Many teachers even teach multiple notations in the same course series, because the different notations highlight different aspects of the subject.

Now, you wrote that your professor's notation was hard to follow, and that you found a proof with an easier-to-follow notation. This sounds as if the second notation was a better fit to the problem space. So, to make your life easier, you could try to pick a school which or a professor who has a reputation for using well-suited formalisms.

$\endgroup$
  • 2
    $\begingroup$ It is pretty obvious that this is not what the OP asked, and it is pretty obvious that you knew that. You just wanted to publish something, even if unrelated to the present discussion... $\endgroup$ – Alex M. Nov 22 '16 at 15:47
  • $\begingroup$ @AlexM. In my opinion, this was not so irrelevant. You are overstating a bit. This made me notice that from a historical point of view, some notations do become easier. $\endgroup$ – polfosol Nov 23 '16 at 8:21
  • $\begingroup$ @Alex M., yes, OP made it clear that "ever" means "during studying", and I thought I made it clear that I understood that. However, the title is what brought me here, and allows a different reading. I am especially interested in the interplay between notation and understanding, and that works over all time scales. To make a slightly stronger connection to OPs intent, I added an example of formalisms that compete both historically and in education. $\endgroup$ – Rainer Blome Nov 24 '16 at 14:43

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.